Tight Gaussian 4-designs

Abstract

A Gaussian t-design is defined as a finite set X in the Euclidean space ℝ n satisfying the condition: 1/V(ℝ n) ∫ℝ n f(x)e -α2∥x∥2 dx=∑ uεX ω(u)f(u) for any polynomial f(x) in n variables of degree at most t, here α is a constant real number and ω is a positive weight function on X. It is easy to see that if X is a Gaussian 2e-design in ℝ n , then |X| ≥ ( en+e). We call X a tight Gaussian 2e-design in ℝ n if |X|=( en+e) holds. In this paper we study tight Gaussian 2e-designs in ℝ n . In particular, we classify tight Gaussian 4-designs in ℝ n with constant weight ω=1/|X| or with weight ω(u)=e-α2∥u∥2/∑xεXe-α2∥x∥2. Moreover we classify tight Gaussian 4-designs in ℝ n on 2 concentric spheres (with arbitrary weight functions). © 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands.